Is the soil adequate for a dry well installation to collect roof runoff

Your eavestroughs (or gutters, in some parts of the world) collect the rain from your roof and direct it through a downspout to the ground level. It sounds like you have concrete splash blocks below your downspouts - like this:

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(Besides preventing soil erosion, the splash blocks keep the dirt from splashing up on your house.)

Drainage is easy - water goes downhill. Just make sure you have a constant downhill path for the water to flow and you're golden. It won't go uphill into a raised planter, unless you have some special capillary mat or plans with long roots. Plantings are good to slow down the flow of water and also absorb it.

You could put any old soil in to fill in the erosion. If you don't have a splashblock, then you should get some -- flat rocks will work also and are often more aesthetically pleasing.

I agree that rain barrels are unsightly, so don't really like them myself. I once put in a french drain, but that was a lot of work. (Dug a trench, put in geotextile, then some gravel on top of the fabric, then 4" weeping tile, then some more gravel, covered the geotextile over top, then backfilled. The geotextile fabric is to prevent dirt from washing into the gravel and eventually blocking it from working effectively for drainage.)

If installed per instructions (buried a minimum of 8") walking on it should be fine. There is also an illustration of adding a vertical pipe in the middle to increase loading (probably a good idea with many of the heavier riding mowers.)